Hyperbolic traveling waves driven by growth

Abstract

We perform the analysis of a hyperbolic model which is the analog of the Fisher-KPP equation. This model accounts for particles that move at maximal speed $\epsilon^{-1}$ ($\epsilon\textgreater{}0$), and proliferate according to a reaction term of monostable type. We study the existence and stability of traveling fronts. We exhibit a transition depending on the parameter $\epsilon$: for small $\epsilon$ the behaviour is essentially the same as for the diffusive Fisher-KPP equation. However, for large $\epsilon$ the traveling front with minimal speed is discontinuous and travels at the maximal speed $\epsilon^{-1}$. The traveling fronts with minimal speed are linearly stable in weighted $L^2$ spaces. We also prove local nonlinear stability of the traveling front with minimal speed when $\epsilon$ is smaller than the transition parameter.